PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE
Nouvelle série, tome 99(113) (2016), 99–108
DOI: 10.2298/PIM1613099M
A NEW MIXED INTEGER LINEAR PROGRAMMING
FORMULATION FOR THE MAXIMUM DEGREE
BOUNDED CONNECTED SUBGRAPH PROBLEM
Zoran Maksimović
Abstract. We give a new mixed integer linear programming (MILP) formulation for Maximum Degree Bounded Connected Subgraph Problem (MDBCSP). The proposed MILP formulation is the first in literature with polynomial number of constraints. Therefore, it will be possible to solve optimally
much more instances before in a reasonable time.
1. Introduction
Let G = (V, E) be an undirected graph, d > 2 an integer, and w : E → R+ a
weight function. The Maximum Degree Bounded Connected Subgraph (MDBCS)
problem is to find a subgraph G′ = (V ′ , E ′ ), where V ′ ⊆ V and E ′ ⊆ E, such
′
that
P the subgraph G is connected, has no vertex with degree exceeding d and
e∈E ′ w(e) has a maximum value. MDBCSP can be illustrated with the following
small example.
Example 1.1. Let |V | = 26, |E| = 29 and graph G = (V, E) with weights is
presented on the left side in Fig. 1.
One optimal solution for d = 2 is presented on the right-hand side in Fig. 1,
with value 99. This result is obtained by using new formulation which is introduced
in Section 4.
It is useful to represent discrete optimization problems as integer programming
problems in order to use different well-known optimization techniques for their
exact solving [8, 11, 22]. The main effort is to design integer linear programming
models with polynomial number of constraints.
2. Related work
MDBCSP is one of the classical NP-hard problems listed in Garey and Johnson’s monograph [10] (as problem GT26), and there is no polynomial factor approximation algorithm for this problem [1]. However, in [6, 7] it is shown that
2010 Mathematics Subject Classification: 90C11; 90C27.
Key words and phrases: integer linear programming, degree-constrained subgraph, combinatorial optimization.
Communicated by Slobodan K. Simić.
99
100
MAKSIMOVIĆ
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Figure 1. Graph G from Example 1 and its optimal solution
when the cost function satisfies triangle inequalities, there are constant factor approximation algorithms for several degree bounded network design problems. The
1
presented approximation algorithm has constant factor 2 + k−1
n + k . The parameterized complexity of some related problems is given in [2]. Note that without
the connectivity constraint, the corresponding problem is known to be solvable in
polynomial time using matching techniques.
There exist graph theoretical results in the literature related to the proposed
problem. For example, in [12] is proved that every 3-connected K1,d -free graph on
n vertices contains a 3-connected spanning subgraph of maximum degree at most
2d − 1. In [13] a similar statement about 2-connected graphs is proved. It is proved
that every 2-connected K1,d -free graph has an a 2-connected spanning subgraph of
maximum degree at most d.
In [23, 24] a research about MDBCSP for chemical compounds is given. Although the problem is NP-hard in a general case, graphs associated with chemical
compounds have polynomial number of spanning trees, so a polynomial time algorithm is proposed.
Subexponential parametrized algorithms on planar graph for finding a connected subgraph with bounded maximum degree and maximal number of edges
(or vertices) is presented in [18, 19]. Their approach uses bidimensionality theory
combined with dynamic programming techniques over branch decompositions of
the input graph, but it is applicable only for small graphs.
A NEW MILP FORMULATION FOR THE MDBCSP
101
Rahman and Kaykobad in [17] proved that a connected graph having independence number d, for d > 2, has a degree-d-bounded spanning tree. They also
presented an O(n2 ) algorithm for constructing such a spanning tree, but independent number of graphs is also an NP-hard problem.
In [5] is presented an O(n)-time algorithm that constructs a connected subgraph of a given triangulation whose maximum degree is at most 14 + ⌈2π/γ⌉, for a
given real number γ, where 0 < γ < π. Sufficient conditions for a connected graph
to have degree bounded trees are presented in [16, 15].
Generalizations of the problem of independent transversals are introduced and
discussed in [21]. It is discussed whether any graph of maximum degree at most
d with a vertex partition into classes of size at least p admits a acyclic H-free
transversal having connected components of order at most r. It is proven that
if the vertex set of a d-regular graph is partitioned into classes of size d + ⌊d/r⌋,
then it is possible to select a transversal inducing vertex disjoin trees on at most
r vertices. Their approach applies appropriate triangulations of the simplex and
SpernerâĂŹs Lemma.
The MDBCSP with maximum diameter is considered in [9]. It discussed some
applications in security, network design and parallel processing, and it derived some
bounds for the order of the largest subgraph in host graphs of practical interest: the
mesh and the hypercube. A heuristic strategy for solving the problem is proposed,
and an approximation ratio for the algorithm is proved. As usual, experimental
results with a variety of host networks showed that the algorithm performs better
in practice than the prediction provided by theoretical approximation ratio.
In [3] is introduced Integer Linear Programming (ILP) formulation with exponential number of constraints, consequently, the standard ILP solvers were able
to solve only very small instances. Additionally, for solving larger instances one
metaheuristic approach based on genetic algorithm was proposed in [3, 4]. It is
based on binary representation and used fine-grained tournament selection, onepoint crossover, simple mutation with frozen genes and caching technique. The
genetic algorithm give approximation solutions on several graphs with up to 212
vertices and 229 edges.
3. An existing ILP formulation
The first integer linear programming (ILP) formulation was proposed in [3].
There are great benefits for representing discrete optimization problems as integer
linear programming models [14, 3] or their generalization [20]. Since it has exponential number of constraints, its ability to solve MDBCSP instances is limited, so
it is unable to solve even the instance presented in Example 1.
ILP formulation uses variables
(
1, e ∈ E ′
(3.1)
xe =
0, e ∈
/ E′
(
1, e ∈ T ′
(3.2)
ye =
0, e ∈
/ T′
102
MAKSIMOVIĆ
zi =
(3.3)
(
1, i ∈ V ′
0, i ∈
/ V′
where e ∈ E, and T ′ is the spanning tree for the subgraph G′ = (V ′ , E ′ ).
Then the MDBCSP was formulated as:
X
(3.4)
max
w(e) · xe
e∈E
subject to:
X
(3.5)
xe 6 d,
i ∈ V,
e:e∋i
(3.6)
ye 6 xe , e ∈ E,
X
X
ye = −1 +
zi
(3.7)
e∈E
i∈V
(3.8)
xe 6 zie ,
e ∈ E,
(3.9)
xe 6 zje ,
e ∈ E,
(3.10)
X
ye 6 |S| − 1,
S ⊆ V, |S| > 3
ie ,je ∈S
(3.11)
xe , ye ∈ {0, 1}, zi ∈ {0, 1},
e ∈ E, i ∈ V
The ie and je denote the vertices that are incident to edge e. As can be seen,
objective function (3.4) maximizes the sum of overall weights, constraints (3.5)
bounded the degree of vertices, (3.6) means that subgraph G′ is connected since it
contains spanning tree given by constraints (3.7)–(3.10). Constraints (3.11) denote
binary nature of decision variables. It is obvious that the number of constraints
(3.10) rise exponentially with the increase of the number of vertices. In order to
produce integer linear programming formulation with polynomial number of constraints, constraints (3.10) should be replaced, because they are the only constraints
from (3.4)–(3.11) whose number is exponential.
Therefore, decision variables y, which, participate in (3.10), should be omitted
together with all constraints referring to these variables.
4. A new MILP formulation
Let us a now construct new ILP formulation based on network flow. The
general idea is to model graph G = (V, E) as network, where each vertex from V ′
consumes amount of flow equal to 1, and the flow can stream only through edges
from E ′ . This idea is completely different from well-known network flow problem,
since V ′ and E ′ and origin of flow are unknown in advance, but the network flow
is only used as general idea to model explanation. Furthermore, the network flow
preservation principle is used, having in mind, that each vertex consumes amount
of flow equal to 1, then the quantity of external flow is equal to cardinality of V ′ .
A NEW MILP FORMULATION FOR THE MDBCSP
103
To accomplish this, we try to input external flow through single vertex (origin of
external network flow), in amount as much as possible, into the network represented
by the graph G = (V, E).
To achieve this goal additional decision variables will be introduced. Binary
decision variables t represent the origin of external network flow, i.e. ti = 1, i ∈ V
if vertex i is the origin of external network flow, and 0 otherwise. The amount of
external network flow is represented by the variables u. Note that ui = 0 for all i,
which are not origin of external network flow. Continuous variables ve denote the
amount of the flow which streams through edge e. Since the graph is undirected,
but network flow must be defined on directed networks, direction of each edge
will be defined in advance. For example, one way of directing graph is to use
lexicographical order (ie < je ). If network flow streams in the direction of edge
e, then ve is positive, otherwise it is negative. Network flow could stream only
through edges from E ′ , while ve = 0, e ∈ E r E ′ . Since flow network is completely
independent with weights of edges (it only guarantees the connectivity of G′ ), then
ve could be equal to 0 even for some e from E ′ .
Let the the binary variables xe and zi be defined as in (3.1) and (3.3), then the
new MILP formulation for the solution of MDBCSP can be formulated as follows
X
(4.1)
max
w(e) · xe
e∈E
subject to:
X
(4.2)
xe 6 d,
i ∈ V,
e:e∋i
(4.3)
xe 6
1
1
zi + zj ,
2 e 2 e
n
X
(4.4)
e∈E
ti = 1
i=1
(4.5)
u i 6 n · ti ,
i∈V
(4.6)
ve 6 n · xe ,
e∈E
(4.7)
ve > −n · xe ,
(4.8)
ui +
X
ve −
e:je =i
(4.9)
zi 6
X
e∈E
ve = zi ,
i∈V
e:ie =i
X
xe
i∈V
e:e∋i
(4.10)
xe ∈ {0, 1}, zi ∈ {0, 1},
ti ∈ [0, +∞), ui ∈ N ∪ {0},
e ∈ E, i ∈ V
ve ∈ [−n, n],
e ∈ E, i ∈ V
104
MAKSIMOVIĆ
Constraints (4.4) and (4.5) implies that external flow in the G′ is obtained from
one origin, with quantity at most n. Constraints (4.6) and (4.7) ensure that the flow
is distributed only by edges in G′ . Network preservation principle is represented
by (4.8). Every vertex from V ′ must have at least one established edge, which is
ensured by (4.9).
Note that, in order to give compact formulation, constraints (3.8) and (3.9)
were substituted with (4.3).
Let us now prove the correctness of the new formulation.
Theorem 4.1. If a subgraph G′ = (V ′ , E ′ ) of a given graph G = (V, E) is the
solution of the Maximum Degree-Bounded Connected Subgraph problem, then the
values of the appropriate variables x, z, t, v, u give the optimal solution of the new
MILP model.
Proof. Let G′ = (V ′ , E ′ ) be an optimal solution of MDBCSP. Let decision
variables x and z be defined as in (3.1) and (3.3), respectively. Let us choose
arbitrary vertex k from V ′ , and define it as origin from which the external flow
streams into the network. Therefore, tk = 1 and uk = |V ′ |, while ti = 0 and ui = 0
for all i 6= k. In order to obtain values of ve the search of graph G′ should be
performed starting from the vertex k (for example we can use depth first search or
breadth first search). Let s(v) be an enumeration of vertices from V ′ generated by
search process. For example, s(k) = 1, for the last searched vertex s is equal to
|V ′ |, while s(v) = 0 for all vertices from V r V ′ . For each edge e ∈ E ′ , the value of
variable ve is defined as follows. If a graph is searched from vertex ie to je through
edge e (s(je ) = s(ie ) + 1), then ve is equal to the number of vertices in the search
subtree rooted by je . Otherwise ve is defined as a number of vertices in the search
subtree rooted by ie multiplied by −1. For all other edges from E ′ and all edges
from E r E ′ it holds that ve = 0.
Since the subgraph G′ has maximal degree of d, then it is obvious that (4.2)
holds. Constraints (4.4) and (4.5) are directly satisfied from the definitions of
variables t and u.
If e ∈ E r E ′ , then from the definition of variables x and v, it holds xe = 0
and ve = 0 and constraints (4.6) and (4.7) are satisfied. In the other case, when
e ∈ E ′ , implying xe = 1. Since ve is defined as number of vertices (or multiplied
by −1), then it must be in [−n, n], and constraints (4.6) and (4.7) are satisfied.
If i ∈ V rV ′ , then all edges e such that ie = i or je = i are in E rE ′ . Therefore,
ui = 0, ve = 0 and zi = 0 and constraint (4.8) is satisfied. If i = k then ui = |V ′ |
and zi = 1. The overall number of vertices in the subtrees of its successors is equal
to the |V ′ | − 1 (all vertices from V ′ are enveloped except vertex k). Therefore, by
the definition of variable v, it holds that
X
X
X
X
X
X
ve
ve −
ve −
ve =
ve +
ve −
e:ie =i
e:je =i
e:ie =k
s(je )<s(k)
e:ie =k
s(je )>s(k)
e:je =k
s(ie )<s(k)
e:je =k
s(ie )>s(k)
Since s(k) = 1, s(ie ) > 1 and s(je ) > 1 then the first and third sums are equal
to zero. Since all elements of the second sum are positive, and all elements in the
A NEW MILP FORMULATION FOR THE MDBCSP
105
fourth sum are negative, overall sum is equal to overall number of vertices in the
subtrees of successors of k i.e. to |V ′ | − 1.
X
X
X
X
X
X
ve −
ve
ve −
ve =
ve +
ve −
e:ie =i
e:je =i
e:ie =i
s(je )<s(i)
e:ie =i
s(je )>s(i)
e:je =i
s(ie )<s(i)
e:je =i
s(ie )>s(i)
In the first and third sum exists only one element whose value is different from
zero. The difference of the first and third sums is equal to the number of vertices
in subtree rooted by i. Since that difference of the second and fourth sums is equal
to the number of vertices in subtrees rooted by successors of i, implying that the
left-hand side of constraint (4.8) is equal to one. For i ∈ V ′ zi = 1, which is the
right-hand side of constraint (4.8).
P
For i ∈ V r V ′ , e ∈ E r E ′ implying zi = 0 and e:e∋i 0 = 0 so constraint (4.9)
is satisfied by default. In the second case when i ∈ V ′ , at least one edge e ∈ E ′ is
incident to the vertex i, i.e. xe = 1, then right-hand side is grater or equal to 1,
which is equal to left-hand side.
Constraints (4.10) are satisfied by the definition of all variables.
For e ∈ E r E ′ constraints (4.3) are trivially satisfied since xe = 0. In the other
case when e ∈ E ′ , then ie , je ∈ E ′ implying xe = zie = zje = 1, which means that
constraints (4.3) are satisfied. P
P
From
P the constraint
P(3.4), e∈E w(e) · xe = e∈E ′ w(e), therefore
max e∈E w(e) · xe > e∈E ′ w(e).
In the last case when i ∈ V ′ and i 6= k, it holds that ui = 0 and zi = 0.
Theorem 4.2. If the values of variables x, z, t, v, u are the optimal solution of
the new MILP model, then the subgraph G′ = (V ′ , E ′ ) where V ′ = {v ∈ V |zv = 1}
and E ′ = {e ∈ E|xe = 1}, is the optimal solution of the MDBCS problem.
Proof. Let values (x, z, t, v, u) be a solution of the MILP model specified
by (4.1) and constraints (4.2)–(4.10). Let us define graph G′ = (V ′ , E ′ ), where
V ′ = {v ∈ V |zv = 1} and E ′ = {e ∈ E|xe = 1}. In continuation it will be proven
that G′ is well defined, connected and d-bounded subgraph of G.
• If e ∈ E ′ then xe = 1. From constraints (4.3) it follows that zie = zje = 1
meaning that both ie , je ∈ V ′ . Therefore, the graph G′ is well defined.
• The subgraph G′ has
P no isolated
P vertices because of (4.9), for each v ∈ V ,
(∀e ∈ E ′ )(v ∈
/ e) ⇒ e:e∋v xe = e:e∋v,e∈ErE ′ xe = 0 ⇒ zv = 0 ⇒ v ∈ V r V ′ .
Directly this means that if v ∈ V ′, there must be at least one e ∈ E ′ incident to v.
• From constraints (4.2) it follows that G′ has no vertex of degree greater than d;
• Now we can prove that V ′ is connected. It is sufficient to prove that for every
partition of V ′ into the two disjoint subsets V1′ , V2′ with |V1′ |, |V2′ | > 2 there exists
at least one edge e ∈ E, which is incident to vertices from both subsets. Suppose,
without loss of generality, that k ∈
/ V1′ .
X
X
X X
ui +
ve −
ve =
zi
i∈V1′
e:je =i
e:ie =i
i∈V1′
106
MAKSIMOVIĆ
The right-hand side R is equal to the |V1′ |. The left-hand side L is equal to
X
X X
X X
ui +
ve −
ve
i∈V1′ e:je =i
i∈V1′
=0+
X
i∈V1′ e:ie =i
e:ie ,je ∈V1′
−
X
X
X
ve −
e:ie ∈V1′ ,je ∈V2′
ve −
e:ie ∈V2′ ,je ∈V1′
Since L = R
ve +
e:ie ∈V2′ ,je ∈V1′
ve −
e:ie ,je ∈V1′
=
X
ve +
X
X
ve
e:je ∈V1′ ,ie ∈V rV ′
X
ve
e:ie ∈V1′ ,je ∈V rV ′
ve
e:ie ∈V1′ ,je ∈V2′
X
ve −
e:ie ∈V2′ ,je ∈V1′
X
ve = |V1′ |,
e:ie ∈V1′ ,je ∈V2′
therefore at least one sum is different from zero and, consequently
(∃e)(((ie ∈ V2′ ∧ je ∈ V1′ ) ∨ (ie ∈ V1′ ∧ je ∈ V2′ ))ve 6= 0.
From that
(∃e)(((ie ∈ V2′ ∧ je ∈ V1′ ) ∨ (ie ∈ V1′ ∧ je ∈ V2′ ))xe = 1,
because if ve > 0, then from (4.6) it is xe = 1. Otherwise, if ve < 0 then from
(4.7) it follows that xe = 1. Therefore, there exists at least one edge e ∈ E,
which is incident to vertices from both subsets V1′ and V2′ .
• For previously defined E ′ it holds that
X
X
w(e) =
w(e) · xe = objMILP (x, z, t, u, v)
e∈E ′
e∈E
Since the optimal solution value of the MDBCSP is maximum of w(G′ ) over
all d-bounded connected subgraphs G′ , and G′ = (V ′ , E ′ ) satisfies all MDBCSP conditions and have objective equal to objMILP (x, z, t, u, v), then the optimal solution value of the MDBCSP on given graph G is greater or equal to
objMILP (x, z, t, u, v).
The nature of variables from presented MILP model can be also illustrated on
Example 1.1.
Let us consider graph from the left side in Figure 1. As was previously mentioned, for d = 2, the optimal solution value is 99. One optimal solution is represented as follows.
Value of t2 = 1 and corresponding u2 = 15 while all others ti = 0 and ui = 0
for i 6= 2.
Values of z-variables are z1 = z2 = z3 = z4 = z5 = z6 = z8 = z10 = z11 =
z15 = z16 = z19 = z21 = z22 = z23 = 1, while z7 = z9 = z12 = z13 = z14 = z17 =
z18 = z20 = z24 = z25 = z26 = 0.
Following x-variables have zero values: x(2,4) = x(5,10) = x(6,7) = x(6,9) =
x(10,12) = x(10,13) = x(10,14) = x(10,16) = x(14,17) = x(15,18) = x(16,20) = x(17,25) =
x(23,24) = x(23,25) = x(23,26) . The corresponding v-variables have also zero values.
A NEW MILP FORMULATION FOR THE MDBCSP
107
All other x-variables have value equal to 1, and the corresponding v-variables have
values: v(1,2) = −6, v(1,3) = 5, v(3,4) = 4, v(4,5) = 3, v(5,6) = 2, v(6,8) = 1,
v(10,11) = −7, v(10,15) = 6, v(15,19) = 5, v(16,21) = 1, v(16,23) = −2, v(19,22) = 4,
v(22,23) = 3, and v(2,11) = 8.
The proposed MILP formulation, using CPLEX 10.1 solver, produced result in
1.119 seconds with 192 constrains, while the CPLEX with existing ILP formulation
failed to produce result with the 3600 seconds time limit with 643568 constraints,
running on the PC with Intel i3 2.2 GHz processor and 4 GB RAM.
5. Conclusions
This paper is devoted to the maximum degree-bounded connected subgraph
problem. We introduced a mixed integer linear programming formulation with
polynomial number of constraints, with proof of its correctness. Therefore the
problem can be solved exactly, by standard ILP solvers, on instances of higher
dimensions than those proposed in the literature.
The future work can be directed in several ways. It would be desirable to investigate the application of some exact method using the proposed MILP formulation.
Further research also should be directed to solving similar problems.
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Department of Natural Sciences and Mathematics
Military Academy
Belgrade
Serbia
[email protected]
(Received 26 04 2013)
(Revised 02 11 2015)